What are examples of problems where neural networks have achieved human-level or higher performance?
Here is an initial list of AI systems that used neural networks and have achieved human-level or superhuman performance. All of these systems are reinforcement learning systems that play videogames.

AlphaGo and AlphaGo Zero (an improved version of AlphaGo that does not use human knowledge but learns by playing against itself) have achieved superhuman performance in the game of go and, in the case of AlphaZero (a generalized version of AlphaGo Zero), also in the games of chess and shogi.
DQN has achieved human-level or superhuman performance in many Atari games
DeepStack has achieved human-level performance in poker
AlphaStar defeated a top professional player in the real-time strategy game StarCraft 2
OpenAI Five defeated world champions in the game of Dota